Long time behavior of fractional impulsive stochastic differential equations with infinite delay

Jiaohui Xu; Tomás Caraballo

doi:10.3934/dcdsb.2018272

Article Contents

2019, Volume 24, Issue 6: 2719-2743. Doi: 10.3934/dcdsb.2018272

This issue Previous Article Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space Next Article H²-stability of some second order fully discrete schemes for the Navier-Stokes equations

Long time behavior of fractional impulsive stochastic differential equations with infinite delay

Jiaohui Xu^1, and
Tomás Caraballo^2,

1.
School of Mathematics and Statistics, Xi’an Jiaotong University Xi’an 710049, China
2.
Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain

Received: March 2018

Revised: May 2018

Early access: October 2018

Published: June 2019

This work has been supported by grant MTM2015-63723-P (MINECO/FEDER, EU) and Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under grant 2010/FQM314, and Proyecto de Excelencia P12-FQM-1492

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Abstract

This paper is first devoted to the local and global existence of mild solutions for a class of fractional impulsive stochastic differential equations with infinite delay driven by both $\mathbb{K}$-valued Q-cylindrical Brownian motion and fractional Brownian motion with Hurst parameter $H∈(1/2,1)$. A general framework which provides an effective way to prove the continuous dependence of mild solutions on initial value is established under some appropriate assumptions. Furthermore, it is also proved the exponential decay to zero of solutions to fractional stochastic impulsive differential equations with infinite delay.

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Mathematics Subject Classification: 34K45, 34G20, 60H10.

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